Vector invariants for the two-dimensional modular representation of the dihedral group of order 2 p

Let F be a field of characteristic p > 2 and D 2 p be the dihedral group of order 2 p . In this paper, we consider the 2-dimensional indecomposable representation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msubsup> </m:math> V₂⁺ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>−</m:mo> </m:mrow> </m:msubsup> </m:math> V₂⁻ of D 2 p and the ring of vector invariants <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>F</m:mi> <m:msup> <m:mrow> <m:mfenced close="]" open="["> <m:mrow> <m:mi>m</m:mi> <m:msubsup> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msubsup> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msup> </m:math> F[mV₂⁺]^D₂ₚ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>F</m:mi> <m:msup> <m:mrow> <m:mfenced close="]" open="["> <m:mrow> <m:mi>m</m:mi> <m:msubsup> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>−</m:mo> </m:mrow> </m:msubsup> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msup> </m:math> F[mV₂⁻]^D₂ₚ for any positive integer m . We give a generating set for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>F</m:mi> <m:msup> <m:mrow> <m:mfenced close="]" open="["> <m:mrow> <m:mi>m</m:mi> <m:msubsup> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msubsup> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msup> </m:math> F[mV₂⁺]^D₂ₚ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>F</m:mi> <m:msup> <m:mrow> <m:mfenced close="]" open="["> <m:mrow> <m:mi>m</m:mi> <m:msubsup> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>−</m:mo> </m:mrow>